Skip to main content
Ch. 7 - Conic Sections
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 7 - Conic SectionsProblem 15
Chapter 8, Problem 15

In Exercises 13–26, use vertices and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. x2/100−y2/64=1

Verified step by step guidance
1
Identify the standard form of the hyperbola equation: x2100 - y264 = 1. Since the x-term is positive and comes first, this is a hyperbola that opens left and right.
Determine the values of a^2 and b^2 from the denominators: a^2 = 100 and b^2 = 64. Then find a = \, \(\text{sqrt}\)(100) = 10 and b = \, \(\text{sqrt}\)(64) = 8.
Locate the vertices on the x-axis at (�� extpm a, 0) = ( extpm 10, 0) because the hyperbola opens horizontally.
Find the foci using the formula c^2 = a^2 + b^2. Calculate c = \, \(\text{sqrt}\)(100 + 64) = \(\text{sqrt}\)(164). The foci are at ( extpm c, 0).
Write the equations of the asymptotes using the slopes extpm rac{b}{a} = extpm rac{8}{10} = extpm rac{4}{5}. The asymptotes pass through the center (0,0), so their equations are y = extpm rac{4}{5}x.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
10m
Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Standard Form of a Hyperbola

A hyperbola in standard form is expressed as (x^2/a^2) - (y^2/b^2) = 1 or (y^2/a^2) - (x^2/b^2) = 1. This form helps identify the orientation of the hyperbola (horizontal or vertical), the vertices located at ±a along the transverse axis, and the relationship between a and b which determines the shape.
Recommended video:
5:50
Asymptotes of Hyperbolas

Foci of a Hyperbola

The foci are two fixed points located along the transverse axis of the hyperbola, found using c^2 = a^2 + b^2. They are essential in defining the hyperbola as the set of points where the difference of distances to the foci is constant. Knowing the foci helps in graphing and understanding the hyperbola's geometry.
Recommended video:
5:22
Foci and Vertices of Hyperbolas

Equations of the Asymptotes

Asymptotes are lines that the hyperbola approaches but never touches. For a hyperbola centered at the origin with horizontal transverse axis, the asymptotes have equations y = ±(b/a)x. These lines guide the shape and direction of the hyperbola branches and are crucial for accurate graphing.
Recommended video:
6:24
Introduction to Asymptotes
Related Practice
Textbook Question

Find the focus and directrix of the parabola with the given equation. Then graph the parabola. 8x2 + 4y = 0

844
views
Textbook Question

Graph the hyperbola. Locate the foci and find the equations of the asymptotes. (x^2)/16 - y^2 = 1

897
views
Textbook Question

Graph each ellipse and locate the foci.4x²+16y² = 64

904
views
Textbook Question

Find the focus and directrix of the parabola with the given equation. Then graph the parabola. y2 - 6x = 0

892
views
Textbook Question

Use vertices and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. x2/9−y2/25=1

837
views
Textbook Question

Graph each ellipse and locate the foci. 25x²+4y² = 100

923
views